100 Essential Things You Didn't Know You Didn't Know (3 page)

Imagine that two attacking players are running (up the page) in parallel straight lines 5 metres apart at a speed of 8 metres per sec towards their opponents’ line. One player, the ‘receiver’, is a metre behind the other, the ‘passer’, who has the ball. The passer throws the ball at 10 metres per sec towards the receiver. The speed of the ball relative to the ground is actually √(10
²
+ 8
²
) = 12.8 metres per sec and it takes a time of 0.4 sec to travel the 5 metres between the players. During this interval the receiver has run a further distance of 8 × 0.4 = 3.2 metres. When the pass was thrown he was 1 metre behind the passer but when he catches the ball he is 2.2 metres in front of him from the point of view of a touch judge standing level with the original pass. He believes there has been a forward pass and waves his flag. But the referee is running alongside the play, doesn’t see the ball go forwards, and so waves play on!

6

Wagons Roll

My heart is like a wheel.

Paul McCartney, ‘Let Me Roll It’

One weekend I noticed that the newspapers were discussing proposals to introduce more restrictive speed limits of 20 mph in built-up areas of the UK and to enforce them with speed cameras wherever possible. Matters of road safety aside, there are some interesting matters of rotational motion that suggest that speed cameras might end up catching large numbers of perplexed cyclists apparently exceeding the speed limit by significant factors. How so?

Suppose that a cycle is moving at speed V towards a speed detector. This means that a wheel hub or the body of the cyclist is moving with speed V with respect to the ground. But look more carefully at what is happening at different points of the spinning wheel. If the wheel doesn’t skid, then the speed of the point of the wheel in contact with the ground must be zero. If the wheel has radius R and is rotating with constant angular velocity Ω revolutions per second, then the speed of the contact point can also be written as V – R Ω. This must be zero and therefore V equals R Ω. The forward speed of the centre of the wheel is V, but the forward speed of the top of the wheel is the sum of V and the rotational speed. This equals V + R Ω and is therefore equal to 2V. If a camera determines the speed of an approaching or receding bicycle by
measuring
the speed of the top of the wheel, then it will register a speed twice as large as the cyclist is moving. An interesting one for m’learned friends perhaps, but I recommend you have a good pair of mudguards.

7

A Sense of Proportion

You can only find truth with logic if you have already found truth without it.

G.K. Chesterton

As you get bigger, you get stronger. We see all sorts of examples of the growth of strength with size in the world around us. The superior strength of heavier boxers, wrestlers and weightlifters is acknowledged by the need to grade competitions by the weight of the participants. But how fast does strength grow with increasing weight or size? Can it keep pace? After all, a small kitten can hold its spiky little tail bolt upright, yet its much bigger mother cannot: her tail bends over under its own weight.

Simple examples can be very illuminating. Take a short breadstick and snap it in half. Now do the same with a much longer one. If you grasped it at the same distance from the snapping point each time you will find that it is no harder to break the long stick than to break the short one. A little reflection shows why this should be so. The stick breaks along a slice through the breadstick. All the action happens there: a thin sheet of molecular bonds in the breadstick is broken and it snaps. The rest of the breadstick is irrelevant. If it were a hundred metres long it wouldn’t make it any harder to break that thin slice of bonds at one point along its length. The strength of the breadstick is given by the number of molecular bonds that have to be broken across its cross-sectional
area
. The bigger that area, the more bonds that need to be broken and the stronger the breadstick. So strength is proportional to cross-sectional area, which is usually proportional to some measure of its diameter squared.

Everyday things like breadsticks and weightlifters have a constant density that is just determined by the average density of the atoms that compose them. But density is proportional to mass divided by volume, which is mass divided by the cube of size. Sitting here on the Earth’s surface, mass is proportional to weight, and so we expect the simple proportionality ‘law’ that for fairly spherical objects

(Strength)
³
∝ (weight)
²

This simple rule of thumb allows us to understand all sorts of things. The ratio of strength to weight is seen to fall as Strength/Weight ∝ (weight)
^-⅓
∝ 1/(size) . So as you grow bigger, your strength does not keep pace with your increasing weight. If all your dimensions expanded uniformly in size, you would eventually be too heavy for your bones to support and you would break. This is why there is a maximum size for land-going structures made of atoms and molecules, whether they are dinosaurs, trees or buildings. Scale them up in shape and size and eventually they will grow so big that their weight is sufficient to sever the molecular bonds at their base and they will collapse under their own weight.

We started by mentioning some sports where the advantage of size and weight is so dramatic that competitors are divided into different classes according to their bodyweight. Our ‘law’ predicts that we should expect to find a straight-line correlation when we plot the cube of the weight lifted against the square of the bodyweight of weightlifters. Here is what happens when you plot that graph for the current men’s world-records in the clean and jerk across the weight categories:

It’s an almost perfect fit! Sometimes mathematics can make life simple. The weightlifter who lies farthest above the line giving the ‘law’ is the strongest lifter ‘pound for pound’, whereas the heaviest lifter, who lifts the largest weight, is actually relatively the weakest when his size is taken into account.

8

Why Does the Other Queue Always Move Faster?

The other man’s grass is always greener.

The sun shines brighter on the other side.

Sung by Petula Clark

You will have noticed that when you join a queue at the airport or the post office, the other queues always seem to move faster. When the traffic is heavy on the motorway, the other lanes always seem to move faster than the one you chose. Even if you change to one of the others, it still goes slower. This situation is often known as ‘Sod’s Law’ and appears to be a manifestation of a deeply antagonistic principle at the heart of reality. Or, perhaps it is merely another manifestation of human paranoia or a selective recording of evidence. We are impressed by coincidences without pausing to recall all the far more numerous non-coincidences we never bothered to keep a note of. In fact, the reason you so often seem to be in the slow queue may not be an illusion. It is a consequence of the fact that on the average you
are
usually in the slow queue!

The reason is simple. On the average, the slow lines and lanes are the ones that have more people and vehicles in them. So, you are more likely to be in those, rather than in the faster moving ones where fewer people are.

The proviso ‘on the average’ is important here. Any particular queue will possess odd features – people who forgot their wallet, have a car that won’t go faster than 30 mph and so on. You won’t invariably be in the slowest line, but
on the average
, when you consider all the lines that you join, you will be more likely to be in the more crowded lines where most people are.

This type of self-selection is a type of bias that can have far-reaching consequences in science and for the analysis of data, especially if it is unnoticed. Suppose you want to determine if people who attend church regularly are healthier than those who do not. There is a pitfall that you have to avoid. The most unhealthy people will not be able to get to church and so just counting heads in the congregation and noting their state of health will give a spurious result. Similarly, when we come to look at the Universe we might have in mind a ‘principle’, inspired by Copernicus, that we must not think that our position in the Universe is special. However, while we should not expect our position to be special in
every
way, it would be a grave mistake to believe that it cannot be special in
any
way. Life may be possible only in places where special conditions exist: it is most likely to be found where there are stars and planets. These structures form in special places where the abundance of dusty raw material is higher than average. So, when we do science or are confronted with data the most important question to ask about the results is always whether some bias is present that leads us preferentially to draw one conclusion rather than another from the evidence.

9

Two’s Company, Three’s a Crowd

What goes up must come down.

Anon.

Two people who get on well together can often find their relationship destabilised by the arrival of a third into their orbit. This is even more noticeable when gravity is the force of attraction involved. Newton taught us that two masses can remain in stable orbit around their centre of mass under their mutual gravitational forces – as do the Earth and the Moon. But if a third body of similar mass is introduced into the system, then something quite dramatic generally happens. One body finds itself kicked out of the system by the gravitational forces, while the two that remain are drawn into a more tightly bound stable orbit.

This simple ‘slingshot’ process is the source of a fantastic counter-intuitive property of Newton’s theory of gravity discovered by Jeff Xia in 1992. First, take four particles of equal mass M and arrange them in two pairs orbiting within two planes that are parallel and with opposite directions of spin so there is no overall rotation. Now introduce a fifth much lighter particle m that oscillates back and forth along the perpendicular through the mass centres of the two pairs. The group of five particles will expand to infinite size in a finite time!

How does this happen? The little oscillating particle runs from one pair to the other, and at the other it creates a little 3-body problem and gets ejected, and the pair recoils outwards to conserve momentum. The lightest particle then travels across to the other pair and the same scenario is repeated. This happens time and time again, without end, and accelerates the two pairs so hard that they become infinitely separated in a finite time, undergoing an infinite number of oscillations in the process.

This example actually solves an old problem posed by philosophers as to whether it is possible to perform an infinite number of actions in a finite time. Clearly, in a Newtonian world where there is no speed limit, it is. Unfortunately (or perhaps fortunately), this behaviour is not possible when Einstein’s relativity is taken into account. No information can be transmitted faster than the speed of light and gravitational forces cannot become arbitrarily strong in Einstein’s theory of motion and gravitation.
Nor
can masses get arbitrarily close to each other and recoil. When two masses of mass M get closer than a distance 4GM/c
²
, where G is Newton’s gravitation constant and c is the speed of light, then a ‘horizon’ surface of no-return forms around them and they form a black hole from which they cannot escape.

The slingshot effect of gravity can be demonstrated in your back garden with a simple experiment. It shows how three bodies can combine to create big recoil as they try to conserve momentum when they pass close to each other (in the case of astronomical bodies) or collide (as it will be in our experiment).

The three bodies will be the Earth, a large ball (like a basket ball or smooth-surfaced football) and a small ball (like a ping-pong or tennis ball). Hold the small ball just above the large ball at about chest height and let them both fall to the ground together. The big ball will hit the ground first and rebound upwards, hitting the small ball while it is still falling. The result is rather dramatic. The small ball bounces up to a height about
nine
times higher than it would have gone if it had just been dropped on the ground from the same height.
^fn1
You might not want to do this indoors!